Bistabilities and annihilation phenomena in electrophysiological cardiac models

Bifurcations and Proarrhythmic Behaviors in Cardiac Electrical Excitations

The heart is a hierarchical dynamic system consisting of molecules, cells, and tissues, and acts as a pump for blood circulation. The pumping function depends critically on the preceding electrical activity, and disturbances in the pattern of excitation propagation lead to cardiac arrhythmia and pump failure. Excitation phenomena in cardiomyocytes have been modeled as a nonlinear dynamical system. Because of the nonlinearity of excitation phenomena, the system dynamics could be complex, and various analyses have been performed to understand the complex dynamics. Understanding the mechanisms underlying proarrhythmic responses in the heart is crucial for developing new ways to prevent and control cardiac arrhythmias and resulting contractile dysfunction. When the heart changes to a pathological state over time, the action potential (AP) in cardiomyocytes may also change to a different state in shape and duration, often undergoing a qualitative change in behavior. Such a dynamic change is called bifurcation. In this review, we first summarize the contribution of ion channels and transporters to AP formation and our knowledge of ion-transport molecules, then briefly describe bifurcation theory for nonlinear dynamical systems, and finally detail its recent progress, focusing on the research that attempts to understand the developing mechanisms of abnormal excitations in cardiomyocytes from the perspective of bifurcation phenomena.

Multiple Dynamical Mechanisms of Phase-2 Early Afterdepolarizations in a Human Ventricular Myocyte Model: Involvement of Spontaneous SR Ca 2+ Release

Early afterdepolarization (EAD) is known to cause lethal ventricular arrhythmias in long QT syndrome (LQTS). In this study, dynamical mechanisms of EAD formation in human ventricular myocytes (HVMs) were investigated using the mathematical model developed by ten Tusscher and Panfilov (Am J Physiol Heart Circ Physiol 291, 2006). We explored how the rapid (IKr) and slow (IKs) components of delayed-rectifier K+ channel currents, L-type Ca2+ channel current (ICa L), Na+/Ca2+ exchanger current (INCX), and intracellular Ca2+ handling via the sarcoplasmic reticulum (SR) contribute to initiation, termination and modulation of phase-2 EADs during pacing in relation to bifurcation phenomena in non-paced model cells. Parameter-dependent dynamical behaviors of the non-paced model cell were determined by calculating stabilities of equilibrium points (EPs) and limit cycles, and bifurcation points to construct bifurcation diagrams. Action potentials (APs) and EADs during pacing were reproduced by numerical simulations for constructing phase diagrams of the paced model cell dynamics. Results are summarized as follows: (1) A modified version of the ten Tusscher-Panfilov model with accelerated ICaL inactivation could reproduce bradycardia-related EADs in LQTS type 2 and β-adrenergic stimulation-induced EADs in LQTS type 1. (2) Two types of EADs with different initiation mechanisms, ICaL reactivation-dependent and spontaneous SR Ca2+ release-mediated EADs, were detected. (3) Termination of EADs (AP repolarization) during pacing depended on the slow activation of IKs. (4) Spontaneous SR Ca2+ releases occurred at higher Ca2+ uptake rates, attributable to the instability of steady-state intracellular Ca2+ concentrations. Dynamical mechanisms of EAD formation and termination in the paced model cell are closely related to stability changes (bifurcations) in dynamical behaviors of the non-paced model cell, but they are model-dependent. Nevertheless, the modified ten Tusscher-Panfilov model would be useful for systematically investigating possible dynamical mechanisms of EAD-related arrhythmias in LQTS.

Propensity for Bistability of Bursting and Silence in the Leech Heart Interneuron

The coexistence of neuronal activity regimes has been reported under normal and pathological conditions. Such multistability could enhance the flexibility of the nervous system and has many implications for motor control, memory, and decision making. Multistability is commonly promoted by neuromodulation targeting specific membrane ionic currents. Here, we investigated how modulation of different ionic currents could affect the neuronal propensity for bistability. We considered a leech heart interneuron model. It exhibits bistability of bursting and silence in a narrow range of the leak current parameters, conductance (g_leak) and reversal potential (E_leak). We assessed the propensity for bistability of the model by using bifurcation diagrams. On the diagram (g_leak, E_leak), we mapped bursting and silent regimes. For the canonical value of E_leak we determined the range of g_leak which supported the bistability. We use this range as an index of propensity for bistability. We investigated how this index was affected by alterations of ionic currents. We systematically changed their conductances, one at a time, and built corresponding bifurcation diagrams in parameter planes of the maximal conductance of a given current and the leak conductance. We found that conductance of only one current substantially affected the index of propensity; the increase of the maximal conductance of the hyperpolarization-activated cationic current increased the propensity index. The second conductance with the strongest effect was the conductance of the low-threshold fast Ca²⁺ current; its reduction increased the propensity index although the effect was about two times smaller in magnitude. Analyzing the model with both changes applied simultaneously, we found that the diagram (g_leak, E_leak) showed a progressively expanded area of bistability of bursting and silence.

Dynamical mechanisms of phase-2 early afterdepolarizations in human ventricular myocytes: insights from bifurcation analyses of two mathematical models

Early afterdepolarization (EAD) is known as a cause of ventricular arrhythmias in long QT syndromes. We theoretically investigated how the rapid (I_Kr) and slow (I_Ks) components of delayed-rectifier K⁺ channel currents, L-type Ca²⁺ channel current (I_CaL), Na⁺/Ca²⁺ exchanger current (I_NCX), Na⁺-K⁺ pump current (I_NaK), intracellular Ca²⁺ (Ca_i) handling via sarcoplasmic reticulum (SR), and intracellular Na⁺ concentration (Na_i) contribute to initiation, termination, and modulation of phase-2 EADs, using two human ventricular myocyte models. Bifurcation structures of dynamical behaviors in model cells were explored by calculating equilibrium points, limit cycles (LCs), and bifurcation points as functions of parameters. EADs were reproduced by numerical simulations. The results are summarized as follows: 1) decreasing I_Ks and/or I_Kr or increasing I_CaL led to EAD generation, to which mid-myocardial cell models were especially susceptible; the parameter regions of EADs overlapped the regions of stable LCs. 2) Two types of EADs (termination mechanisms), I_Ks activation-dependent and I_CaL inactivation-dependent EADs, were detected; I_Ks was not necessarily required for EAD formation. 3) Inhibiting I_NCX suppressed EADs via facilitating Ca²⁺-dependent I_CaL inactivation. 4) Ca_i dynamics (SR Ca²⁺ handling) and Na_i strongly affected bifurcations and EAD generation in model cells via modulating I_CaL, I_NCX, and I_NaK Parameter regions of EADs, often overlapping those of stable LCs, shifted depending on Ca_i and Na_i in stationary and dynamic states. 5) Bradycardia-related induction of EADs was mainly due to decreases in Na_i at lower pacing rates. This study demonstrates that bifurcation analysis allows us to understand the dynamical mechanisms of EAD formation more profoundly.

NEW & NOTEWORTHY

We investigated mechanisms of phase-2 early afterdepolarization (EAD) by bifurcation analyses of human ventricular myocyte (HVM) models. EAD formation in paced HVMs basically depended on bifurcation phenomena in non-paced HVMs, but was strongly affected by intracellular ion concentrations in stationary and dynamic states. EAD generation did not necessarily require I_Ks.

Effect of hyperpolarization-activated current I(f) on robustness of sinoatrial node pacemaking: theoretical study on influence of intracellular Na(+) concentration

To elucidate the effects of hyperpolarization-activated current I(f) on robustness of sinoatrial node (SAN) pacemaking in connection with intracellular Na(+) concentration (Na(i)) changes, we theoretically investigated 1) the impacts of I(f) on dynamical properties of SAN model cells during inhibition of L-type Ca(2+) channel currents (I(CaL)) or hyperpolarizing loads and 2) I(f)-dependent changes in Na(i) and their effects on dynamical properties of model cells. Bifurcation analyses were performed for Na(i)-variable and Na(i)-fixed versions of mathematical models for rabbit SAN cells; equilibrium points (EPs), limit cycles (LCs), and their stability were determined as functions of model parameters. Increasing I(f) conductance (g(f)) shrank I(CaL) conductance (g(CaL)) regions of unstable EPs and stable LCs (rhythmic firings) in the Na(i)-variable system but slightly broadened that of rhythmic firings at lower g(f) in the Na(i)-fixed system. In the Na(i)-variable system, increased g(f) yielded elevations in Na(i) at EPs and during spontaneous oscillations, which caused EP stabilization and shrinkage in the parameter regions of unstable EPs and rhythmic firings. As g(f) increased, parameter regions of unstable EPs and stable LCs determined for hyperpolarizing loads shrank in the Na(i)-variable system but were enlarged in the Na(i)-fixed system. These findings suggest that 1) I(f) does not enhance but rather attenuates robustness of rabbit SAN cells via facilitating EP stabilization and LC destabilization even in physiological g(f) ranges; and 2) the enhancing effect of I(f) on robustness of pacemaker activity, which could be observed at lower g(f) when Na(i) was fixed, is actually reversed by I(f)-dependent changes in Na(i).

Dynamics of early afterdepolarization-mediated triggered activity in cardiac monolayers

Early afterdepolarizations (EADs) are voltage oscillations that occur during the repolarizing phase of the cardiac action potential and cause cardiac arrhythmias in a variety of clinical settings. EADs occur in the setting of reduced repolarization reserve and increased inward-over-outward currents, which intuitively explains the repolarization delay but does not mechanistically explain the time-dependent voltage oscillations that are characteristic of EADs. In a recent theoretical study, we identified a dual Hopf-homoclinic bifurcation as a dynamical mechanism that causes voltage oscillations during EADs, depending on the amplitude and kinetics of the L-type Ca(2+) channel (LTCC) current relative to the repolarizing K(+) currents. Here we demonstrate this mechanism experimentally. We show that cardiac monolayers exposed to the LTCC agonists BayK8644 and isoproterenol produce EAD bursts that are suppressed by the LTCC blocker nitrendipine but not by the Na(+) current blocker tetrodoxin, depletion of intracellular Ca(2+) stores with thapsigargin and caffeine, or buffering of intracellular Ca(2+) with BAPTA-AM. These EAD bursts exhibited a key dynamical signature of the dual Hopf-homoclinic bifurcation mechanism, namely, a gradual slowing in the frequency of oscillations before burst termination. A detailed cardiac action potential model reproduced the experimental observations, and identified intracellular Na(+) accumulation as the likely mechanism for terminating EAD bursts. Our findings in cardiac monolayers provide direct support for the Hopf-homoclinic bifurcation mechanism of EAD-mediated triggered activity, and raise the possibility that this mechanism may also contribute to EAD formation in clinical settings such as long QT syndromes, heart failure, and increased sympathetic output.

Roles of sarcoplasmic reticulum Ca2+ cycling and Na+/Ca2+ exchanger in sinoatrial node pacemaking: Insights from bifurcation analysis of mathematical models

To elucidate the roles of sarcoplasmic reticulum (SR) Ca2+ cycling and Na+/Ca2+ exchanger (NCX) in sinoatrial node (SAN) pacemaking, we have applied stability and bifurcation analyses to a coupled-clock system model developed by Maltsev and Lakatta ( Am J Physiol Heart Circ Physiol 296: H594-H615, 2009). Equilibrium point (EP) at which the system is stationary (i.e., the oscillatory system fails to function), periodic orbit (limit cycle), and their stability were determined as functions of model parameters. The stability analysis to detect bifurcation points confirmed crucial importance of SR Ca2+ pumping rate constant ( Pup), NCX density ( kNCX), and L-type Ca2+ channel conductance for the system function reported in previous parameter-dependent numerical simulations. We showed, however, that the model cell does not exhibit self-sustained automaticity of SR Ca2+ release at any clamped voltage and therefore needs further tuning to reproduce oscillatory local Ca2+ release and net membrane current reported experimentally at −10 mV. Our further extended bifurcation analyses revealed important novel features of the pacemaker system that go beyond prior numerical simulations in relation to the roles of SR Ca2+ cycling and NCX in SAN pacemaking. Specifically, we found that 1) NCX contributes to EP instability and enhancement of robustness in the full system during normal spontaneous action potential firings, while stabilizing EPs to prevent sustained Ca2+ oscillations under voltage clamping; 2) SR requires relatively large kNCX and subsarcolemmal Ca2+ diffusion barrier (i.e., subspace) to contribute to EP destabilization and enhancement of robustness; and 3) decrementing Pup or kNCX decreased the full system robustness against hyperpolarizing loads because EP stabilization and cessation of pacemaking were observed at the lower critical amplitude of hyperpolarizing bias currents, suggesting that SR Ca2+ cycling contributes to enhancement of the full system robustness by modulating NCX currents and promoting EP destabilization.

Roles of hyperpolarization-activated current If in sinoatrial node pacemaking: insights from bifurcation analysis of mathematical models

To elucidate the roles of hyperpolarization-activated current (I(f)) in sinoatrial node (SAN) pacemaking, we theoretically investigated 1) the effects of I(f) on stability and bifurcation during hyperpolarization of SAN cells; 2) combined effects of I(f) and the sustained inward current (I(st)) or Na(+) channel current (I(Na)) on robustness of pacemaking against hyperpolarization; and 3) whether blocking I(f) abolishes pacemaker activity under certain conditions. Bifurcation analyses were performed for mathematical models of rabbit SAN cells; equilibrium points (EPs), periodic orbits, and their stability were determined as functions of parameters. Unstable steady-state potential region determined with applications of constant bias currents shrunk as I(f) density increased. In the central SAN cell, the critical acetylcholine concentration at which bifurcations, to yield a stable EP and quiescence, occur was increased by smaller I(f), but decreased by larger I(f). In contrast, the critical acetylcholine concentration and conductance of gap junctions between SAN and atrial cells at bifurcations progressively increased with enhancing I(f) in the peripheral SAN cell. These effects of I(f) were significantly attenuated by eliminating I(st) or I(Na), or by accelerating their inactivation. Under hyperpolarized conditions, blocking I(f) abolished SAN pacemaking via bifurcations. These results suggest that 1) I(f) itself cannot destabilize EPs; 2) I(f) improves SAN cell robustness against parasympathetic stimulation via preventing bifurcations in the presence of I(st) or I(Na); 3) I(f) dramatically enhances peripheral cell robustness against electrotonic loads of the atrium in combination with I(Na); and 4) pacemaker activity of hyperpolarized SAN cells could be abolished by blocking I(f).

Regional difference in dynamical property of sinoatrial node pacemaking: role of na+ channel current

To elucidate the regional differences in sinoatrial node pacemaking mechanisms, we investigated 1), bifurcation structures during current blocks or hyperpolarization of the central and peripheral cells, 2), ionic bases of regional differences in bifurcation structures, and 3), the role of Na(+) channel current (I(Na)) in peripheral cell pacemaking. Bifurcation analyses were performed for mathematical models of the rabbit sinoatrial node central and peripheral cells; equilibrium points, periodic orbits, and their stability were determined as functions of parameters. Structural stability against applications of acetylcholine or electrotonic modulations of the atrium was also evaluated. Blocking L-type Ca(2+) channel current (I(Ca,L)) stabilized equilibrium points and abolished pacemaking in both the center and periphery. Critical acetylcholine concentration and gap junction conductance for pacemaker cessation were higher in the periphery than in the center, being dramatically reduced by blocking I(Na). Under hyperpolarized conditions, blocking I(Na), but not eliminating I(Ca,L), abolished peripheral cell pacemaking. These results suggest that 1), I(Ca,L) is responsible for basal pacemaking in both the central and peripheral cells, 2), the peripheral cell is more robust in withstanding hyperpolarizing loads than the central cell, 3), I(Na) improves the structural stability to hyperpolarizing loads, and 4), I(Na)-dependent pacemaking is possible in hyperpolarized peripheral cells.

Effects of pacemaker currents on creation and modulation of human ventricular pacemaker: theoretical study with application to biological pacemaker engineering

A cardiac biological pacemaker (BP) has been created by suppression of the inward rectifier K+ current ( IK1) or overexpression of the hyperpolarization-activated current ( Ih). We theoretically investigated the effects of incorporating Ih, T-type Ca2+ current ( ICa,T), sustained inward current ( Ist), and/or low-voltage-activated L-type Ca2+ channel current ( ICa,LD) on 1) creation of BP cells, 2) robustness of BP activity to electrotonic loads of nonpacemaking (NP) cells, and 3) BP cell ability to drive NP cells. We used a single-cell model for human ventricular myocytes (HVMs) and also coupled-cell models composed of BP and NP cells. Bifurcation structures of the model cells were explored during changes in conductance of the currents and gap junction. Incorporating the pacemaker currents did not yield BP activity in HVM with normal IK1 but increased the critical IK1 conductance for BP activity to emerge. Expressing Ih appeared to be most helpful in facilitating creation of BP cells via IK1 suppression. In the coupled-cell model, Ist significantly enlarged the gap conductance ( GC) region where stable BP cell pacemaking and NP cell driving occur, reducing the number of BP cells required for robust pacemaking and driving. In contrast, Ih enlarged the GC region of pacemaking and driving only when IK1 of the NP cell was relatively low. ICa,T or ICa,LD exerted effects similar to those of Ist but caused shrinkage or irregularity of BP oscillations. These findings suggest that expressing Ist most effectively improves the structural stability of BPs to electrotonic loads and the BP ability to drive the ventricle.

Dynamical mechanisms of pacemaker generation in IK1-downregulated human ventricular myocytes: insights from bifurcation analyses of a mathematical model

Dynamical mechanisms of the biological pacemaker (BP) generation in human ventricular myocytes were investigated by bifurcation analyses of a mathematical model. Equilibrium points (EPs), periodic orbits, stability of EPs, and bifurcation points were determined as functions of bifurcation parameters, such as the maximum conductance of inward-rectifier K+ current (I(K1)), for constructing bifurcation diagrams. Stable limit cycles (BP activity) abruptly appeared around an unstable EP via a saddle-node bifurcation when I(K1) was suppressed by 84.6%. After the bifurcation at which a stable EP disappears, the I(K1)-reduced system has an unstable EP only, which is essentially important for stable pacemaking. To elucidate how individual sarcolemmal currents contribute to EP instability and BP generation, we further explored the bifurcation structures of the system during changes in L-type Ca2+ channel current (I(Ca,L)), delayed-rectifier K+ currents (I(K)), or Na(+)/Ca2+ exchanger current (I(NaCa)). Our results suggest that 1), I(Ca,L) is, but I(K) or I(NaCa) is not, responsible for EP instability as a requisite to stable BP generation; 2), I(K) is indispensable for robust pacemaking with large amplitude, high upstroke velocity, and stable frequency; and 3), I(NaCa) is the dominant pacemaker current but is not necessarily required for the generation of spontaneous oscillations.

Roles of L-type Ca2+ and delayed-rectifier K+ currents in sinoatrial node pacemaking: insights from stability and bifurcation analyses of a mathematical model

To elucidate the dynamical mechanisms of the sinoatrial (SA) node pacemaker activity, we investigated the roles of L-type Ca2+ (ICa,L) and delayed-rectifier K+ (IKr) currents in pacemaking by stability and bifurcation analyses of our rabbit SA node model (Kurata Y, Hisatome I, Imanishi S, and Shibamoto T. Am J Physiol Heart Circ Physiol 283: H2074-H2101, 2002). Equilibrium points (EPs), periodic orbits, stability of EPs, and Hopf bifurcation points were calculated as functions of conductance or gating time constants of the currents for constructing bifurcation diagrams. Structural stability (robustness) of the system was also evaluated by computing stability and dynamics during applications of constant bias currents (Ibias). Blocking ICa,L or IKr caused stabilization of an EP and cessation of pacemaking via a Hopf bifurcation. The unstable zero-current potential region determined with Ibias applications, where spontaneous oscillations appear, shrunk and finally disappeared as ICa,L diminished, but shrunk little when IKr was eliminated. The reduced system, including no time-dependent current except ICa,L, exhibited pacemaker activity. These results suggest that ICa,L is responsible for EP instability and pacemaker generation, whereas IKr is not necessarily required for constructing a pacemaker cell system. We further explored the effects of various K+ currents with different kinetics on stability and dynamics of the model cell. The original IKr of delayed activation and inward rectification appeared to be most favorable for generating large-amplitude oscillations with stable frequency, suggesting that IKr acts as an oscillation amplifier and frequency stabilizer. IKr may also play an important role in preventing bifurcation to quiescence of the system.

Limit cycle oscillations in pacemaker cells

In recent decades, several mathematical models describing the pacemaker activity of the rabbit sinoatrial node have been developed. We observe that membrane voltage and membrane charge are treated as independent dynamic variables in some of those models, resulting in an infinite number of limit cycles. Then we display numerical results from a new model where membrane voltage (v) is not a dynamic variable, and observe a limit cycle oscillation that can be reached from many different initial conditions.

Studies on re-entrant arrhythmias and ectopic beats in excitable tissues by bifurcation analyses

A phase-plane bifurcation analysis is a useful way to theoretically understand how various types of arrhythmias may arise from excitable tissues. In this paper, we have performed phase-plane bifurcation analysis to characterize arrhythmogenic states in excitable tissues. To achieve this, we have first formulated a model which is simple enough to be mathematically tractable, yet captures the non-linear features of cardiac excitation and conduction. In this model, single cells are connected in a circular fashion by gap conductances. Each cell carries the following two types of currents: a passive outward current and an inward "excitable" current which contains an activation and an inactivation gate. The activation gate is responsible for the upstroke of action potential and inactivation gate is responsible for the termination of the plateau potential. With this model, we have constructed bifurcation diagrams as a function of a bifurcation parameter. The parameter chosen as the bifurcation parameter has the property of raising maximum diastolic potential while shorting the refractory period. Our analysis revealed the existence of three distinct multi-stable phases in certain ranges of the bifurcation parameter: (1) bistability between a rotor and a quiescent state, (2) bistability between rotor and ectopic beats, and (3) three stable states co-existing among quiescent state, rotor, and ectopic beats. In these three regions, external impulses exert very distinct effects: In region 1, a brief current pulse can annihilate a re-entrant arrhythmia to quiescence. To initiate re-entry from a quiescent tissue, however, it takes two pulses (a primary pulse followed by a premature pulse at a site different from the "primary" site). In region 2, a brief pulse can convert a re-entrant arrhythmia to ectopic beats. To convert the ectopic beats back to circus movement, these beats have to be suppressed by a few brief current pulses to initiate one-way propagation. Depending on the frequency and strength of impulses in region 3, the tissue can switch back and forth among quiescence, circus movement, and ectopic beats. For comparison, we have also included a more complete Beeler-Reuter cardiac cell model in our analysis and obtained essentially the same results. From the behavioral similarities of these models, we conclude that re-entrant and ectopic arrhythmias must be intrinsic properties of excitable tissues and external stimuli can convert one mode of arrhythmia to another in the multistability regions.(ABSTRACT TRUNCATED AT 400 WORDS)